A random matrix formulation of fidelity decay

Abstract

We propose to study echo dynamics in a random matrix framework, where we assume that the perturbation is time independent, random and orthogonally invariant. This allows to use a basis in which the unperturbed Hamiltonian is diagonal and its properties are thus largely determined by its spectral statistics. We concentrate on the effect of spectral correlations usually associated to chaos and disregard secular variations in spectral density. We obtain analytic results for the fidelity decay in the linear response regime. To extend the domain of validity, we heuristically exponentiate the linear response result. The resulting expressions, exact in the perturbative limit, are accurate approximations in the transition region between the ``Fermi golden rule'' and the perturbative regimes, as examplarily verified for a deterministic chaotic system. To sense the effect of spectral stiffness, we apply our model also to the extreme cases of random spectra and equidistant spectra. In our analytical approximations as well as in extensive Monte Carlo calculations, we find that fidelity decay is fastest for random spectra and slowest for equidistant ones, while the classical ensembles lie in between. We conclude that spectral stiffness systematically enhances fidelity.